Negative c-axis magnetoresistance in graphite
Y. Kopelevich1, 2, R. R. da Silva1, J. C. Medina Pantoja1, and A. M. Bratkovsky2
1
Instituto de Física “Gleb Wataghin“, Universidade Estadual de Campinas, UNICAMP 13083970, Campinas, São Paulo, Brasil
2
Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, California 94304
ABSTRACT
We have studied the c-axis interlayer magnetoresistance (ILMR), Rc(B) in graphite. The
measurements have been performed on strongly anisotropic highly oriented pyrolytic graphite
(HOPG) samples in magnetic field up to B = 9 T applied both parallel and perpendicular to the
sample c-axis in the temperature interval 2 K  T  300 K. We have observed negative
magnetoresistance, dRc/dB < 0, for B || c-axis above a certain field Bm(T) that reaches its
minimum value Bm = 5.4 T at T = 150 K. The results can be consistently understood assuming
that ILMR is related to a tunneling between zero-energy Landau levels of quasi-two-dimensional
Dirac fermions, in a close analogy with the behavior reported for -(BEDT-TTF)2I3 [N. Tajima
et al., Phys. Rev. Lett. 102, 176403 (2009)], another multilayer Dirac electron system.
PACS numbers: 72.15.Gd, 71.70.Di, 73.22.Pr, 73.21.Ac
1
Graphite consists of N > 1 layers of carbon atoms packed in honeycomb lattice, dubbed
graphenes, with two non-equivalent sites, A and B, in the Bernal (ABAB...) stacking
configuration [1]. Conducting (conjugated) -electrons move within planes, formed by pz-wave
functions density maxima located parallel (above and below) to graphene layers. In the absence
of interlayer electron hopping, charge-carrying quasi-particles (QP) have the linear dispersion
relation E(p) = v|p|, and the Fermi surface is reduced to two points (K and K′) at the opposite
corners of the 2D hexagonal Brillouin zone. Such carriers can be described as massless (2+1)D
Dirac fermions (DF) [2] providing a link to relativistic models for particles with an effective
“light” velocity v ≈ 106 ms-1. On the other hand, according to Slonczewski-Weiss-McClure
(SWMC) model [1], the interlayer coupling leads to a dispersion in the pz-direction with cigarlike electron (K) and hole (H) Fermi surface pockets elongated along the corner edge HKH of a
3D Brillouin zone, and no linear dispersion is expected, except very close to the H point [3].
However, studies of Shubnikov de Haas (SdH) and de Haas van Alphen (dHvA) quantum
oscillations [4] showed that DF in graphite occupied an unexpectedly large phase volume. The
DFs in graphite were also detected by means of angle-resolved photoemission spectroscopy
(ARPES) [5], scanning-tunneling-spectroscopy (STS) [6], and infrared magneto-transmission [7]
techniques. Besides, micro-Raman [8], STS [9] and microwave magneto-absorption [10]
measurements provided evidence for the existence of independent (decoupled) graphene layers
in bulk graphite.
The question whether graphene layers are situated only at the sample surface or they are
distributed through the sample volume remains unclear. Aiming to verify the possible multigraphene behavior of graphite, we have studied in the present work the c-axis interlayer
magnetoresistance (ILMR). In particular, it is expected that ILMR is dominated by interlayer
2
tunneling between zero-energy Landau levels of quasi-2D DF, and the perpendicular magnetic
field, increasing the zero energy state degeneracy, results in the resistance decreasing with the
field, i. e. negative ILMR (NILMR) [11]. Here we report the observation of NILMR in our most
anisotropic graphite samples that resembles the resistance behavior reported for -(BEDTTTF)2I3 [12], another multilayer Dirac electron system.
The measurements were performed on thoroughly characterized [13] HOPG-UC (Union
Carbide Co.) with the room temperature, zero-field, out-of-plane/basal-plane resistivity ratio
c/b = 3·104, c = 0.1 cm, and mosaicity of 0.4° (FWHM obtained from x-ray rocking curves).
X-ray diffraction (-2) spectra revealed a characteristic hexagonal graphite structure in the
Bernal (ABAB…) stacking configuration, with no signature of the rhombohedral phase. The
obtained crystal lattice parameters are a = 2.48 Å and c = 6.71 Å. Both dc and low-frequency (f =
1 Hz) ac resistance measurements were made using commercial He-4, B = 9 T cryostats. For the
measurements, silver paste electrodes were placed on the sample surface(s). The c-axis resistance
Rc(B,T) was measured by attaching two electrodes to each of the main (basal) sample surfaces;
one is point-like in the middle and other surrounding it and contacted on the rest surface,
assuring a uniform current distribution. Complementary measurements of the in-plane resistance
Rb(B, T) were performed by attaching four contacts to the same sample surface. Measurements
were performed for both B || c and B  c configurations. The results presented below were
obtained for the sample with dimensions l x w x t = 5 x 5 x 1 mm3 (t || c-axis).
Figure 1 (a, b) presents Rc(B) isotherms recorded for various temperatures. The salient
feature of the data, i. e. the maxima in Rc(B), is indicated by arrows. The negative
magnetoresistance (dRc/dB < 0) reveals itself at B > Bm(T). As shown in Fig. 2, Bm(T) is a nonmonotonic function having the minimum at T ~ 150 K.
3
Taking into account that the basal-plane MR in well graphitized samples is positive and
very big [1], see also the inset in Fig. 1(a), the occurrence of negative MR in the certain domain
of the B-T plane (Fig. 2) should be specific to the c-axis transport. In order to clarify its origin,
we first turn our attention to the zero-field anisotropic electrical transport.
Figure 3 presents normalized c(T) and b(T) resistivity curves obtained for B = 0. As
can be seen from Fig. 3, c(T) demonstrates insulating-like (dc/dT < 0) behavior for T > 50 K,
whereas b(T) rapidly decreases for T < 150 K (db/dT > 0). As a result, the anisotropy c/b
increases as the temperature decreases (see the inset in Fig. 3), representing the characteristic
feature of most anisotropic HOPG. The ratio c/b vs. T can be best fitted by the equation:
c/b = a + bT-,
(1)
where  = 0.5. The insulating-type c(T) is the inherent property of a disorder-free graphite, but
the presence of lattice defects can act as short-circuits between the graphene planes, reducing the
measured c(T) compared to its true value [1]. Recent theoretical studies of the perpendicular
transport in graphene multi-layers [14] led to similar conclusions: (i) the perpendicular transport
is enhanced by structural disorder; (ii) the cleaner the system the larger the anisotropy; (iii) the
resistivity ratio c/b ~ T- at the Dirac point, and diverges when T  0. However, the saturation
of c/b with the temperature lowering is expected for finite doping (chemical potential), being in
agreement with our observations (see the inset in Fig. 3). The predicted exponent  = 0.66(6) is
also close to the experimental value  = 0.5 found in the present work.
Because of the structural disorder, the current in the measurements flows not only
perpendicular but also parallel to graphene planes. This explains the occurrence of SdH
4
oscillations in the B || I || c configuration, clearly seen for T = 2 K and T = 10 K in Fig. 1(a), top
curve, as well as the metallic-like behavior of Rc(T) at T < 50 K (Fig. 3): both effects are likely
governed by Rb(T).
Then, the resistance measured along the c-axis can be described by the equation for two
parallel resistors Reff = RcRb/(Rc+Rb), where Rb(B) = R0 + aBs(T) [1, 15] and
Rc(B) = A/(B + B0),
(2)
predicted for multilayer DF systems [11, 12], where A and B0 are fitting parameters. The
background physics behind the decreasing Rc vs. B (NILMR) is the Dirac-type Landau level
quantization En =  (2evF2|n|B)1/2 [2], where the lowest Landau level is located precisely at E0 =
0 (zero mode). This has an important consequence: when the interlayer transport is governed by
the tunneling between zero-energy Landau levels, the increase of zero-mode degeneracy with the
field leads to NILMR [11, 12]. Red lines in Fig. 1 are obtained from the equation for Reff(B)
exemplifying good agreement with the experimental results. Somewhat similar interpretation of
the field-driven crossover from positive to negative MR has recently been proposed in the
context of -(BEDT-TTF)2I3 [16], assuming a non-vertical interlayer tunneling. According to
Ref. [16], the crossover field Bm that marks a maximum in Rc(B) corresponds to the crossover
from the inter-Landau level mixing regime (B  Bm) to the interlayer tunneling regime between
well separated zero-energy Landau levels (B  Bm). This implies that for B > Bm, one has E1 - E0
> kBT, , tc where  = ħ/ ( is the transport relaxation time) and tc measure the strength of a
quenched disorder and interlayer tunneling, respectively. It is worth noting, that the old SWMC
model for graphite suggests tc ≈ 0.39 eV >> kBT, and at first glance the comparison of our
5
experimental results with the theoretical models [11, 16], where tc < kBT, may not be appropriate.
However, this value of tc is nearly two orders of magnitude larger than the value ~ 5 meV
reported e. g. by Haering and Wallace [17] who pointed out the 2D character of QPs in graphite,
see also Refs. [4, 18]. Also, recent measurements [19] of current-voltage (I-V) characteristics
performed on graphitic mesas suggested the interlayer tunneling of Dirac fermions between
Landau levels. It seems, both energy scales are relevant in graphite which can be viewed as the
stack of alternating “blocks” of strongly and weakly coupled graphene planes [18, 20], so that
the NILMR originates from the tunneling between nearly decoupled graphitic planes. Taking
characteristic (T) for graphite [21], the inequality (E1 - E0)/kB[K] ≈ ±420|n|1/2(B[T])1/2 > {T, }
is satisfied over the whole NILMR domain on the B-T plane in Fig. 2.
From this perspective, one also understands the non-monotonic Bm(T). As the
temperature decreases from 300 K to ~ 100 K, the condition E1 - E0 > {kBT, } improves and
hence Bm decreases. At the same time, the inequality tc < kBT inevitably inverts as T  0,
implying the divergence of Bm with the temperature lowering as Fig. 2 illustrates. Assuming that
the minimum in Bm(T) corresponds to the condition tc  kBT, one gets tc ~ 10 meV, close to the
value obtained in Ref. [17].
Testing further the theoretical model for NILMR [11], we performed Rc(B)
measurements with B || basal planes (Fig. 4). As Fig. 4 illustrates, in this geometry the negative
MR does not occur for all studied temperatures and magnetic fields, providing evidence that
NILMR is indeed associated with the field component perpendicular to basal planes.
Summarizing, we report the first observation of negative interlayer magnetoresistance
(NILMR) in strongly anisotropic highly ordered graphite that can be consistently understood
within a framework of tunneling models between zero-energy Landau levels of quasi-two-
6
dimensional Dirac fermions [11, 16]. This finding together with the zero-field resistivity
anisotropy measurements, performed in this work, provides an additional experimental evidence
that graphite consists (at least partially ) of weakly coupled graphene planes with the Dirac-type
quasiparticle spectrum.
This work was partially supported by FAPESP, CNPq, and INCT NAMITEC.
7
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9
FIGURE CAPTIONS
Fig.1. (Color online) Rc(B) isotherms recorded for various temperatures. The negative
magnetoresistance (dRc/dB < 0) takes place for B > Bm(T) indicated by arrow. Red lines are
obtained from the equation Reff = RcRb/(Rc+Rb), where Rb(B) = R0 + aBs(T) is the basal-plane
resistance contribution (see text) and Rc(B) = A/(B + B0) with R0, a, A, and B0 being fitting
parameters: R0 = 0.15 , a = 0.26 ·T-1.27, A = 19 ·T, B0 = 10 T, s = 1.3 (T = 50 K); R0 =
0.085 , a = 0.1 ·T-1.5, A = 4.85 ·T, B0 = 10 T, s = 1.5 (T = 100 K); R0 = 0.08 , a = 0.04
·T-1.6, A = 2.5 ·T, B0 = 11 T, s = 1.6 (T = 175 K). The inset in (a) exemplifies Rb(B)/Rb(0)
measured with the current flowing parallel to basal graphitic planes.
Fig.2. Bm(T) separates positive (dRc/dB > 0) and negative (dRc/dB < 0) magnetoresistance.
Fig.3. (Color online) Normalized c-axis c(T)/c(300 K) and basal-plane b(T)/b(300 K)
resistivity curves obtained for B = 0. The inset presents the anisotropy c/b vs. T; the red line is
obtained from Eq. (1): a = 1.1, b = 40 K,  = 0.5.
Fig.4. Rc(B) measured at various temperatures and B || basal planes.
10
Fig. 1
2K
4
Rb(B)/R(0)
100
(b)
100 K
100 K
50
0
0 2 4 6 8
Bmax(T)
10 K
125 K
0.2
R c ( )
20 K
150 K
2
175 K
30 K
250 K
40 K
50 K
300 K
(a)
0
0
2
4
6
8
0
B (T)
11
2
4
6
8
0.1
Fig. 2
Bm (T)
8
NEGATIVE
MAGNETORESISTANCE
7
6
5
50
100
150
T (K)
12
200
250
300
Fig. 3
2.0
B=0
c(T)
b(T)
1.0
15
10
c/b = a +bT
-
-4
10 c/b
(T)/(300 K)
1.5
0.5
0
50
5
100
150
T (K)
13
200
250
300
Fig. 4
T=2K
1.0
B || basal planes
20 K
0.8
R c ( )
10 K
30 K
0.6
40 K
0.4
50 K
70 K
0.2
0.0
100 K
150 K
200 K
250 K
0
2
4
6
B (T)
14
8
10